# A Time-spectral Approach to Numerical Weather Prediction

## TL;DR

This paper explores a time-spectral method called GWRM for numerical weather prediction, demonstrating it can be more accurate and significantly faster than traditional finite difference methods, especially at high accuracy levels.

## Contribution

It introduces the GWRM time-spectral algorithm for NWP, showing its advantages over finite difference methods in accuracy, efficiency, and analytical solution generation.

## Key findings

- GWRM achieves comparable accuracy to explicit and implicit methods.
- GWRM is up to four times faster in perturbative scenarios.
- GWRM allows larger time intervals, reducing computational cost.

## Abstract

Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal, CFL-like critera are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM. Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling.